Band spectrum singularities for Schr\"odinger operators

Abstract

In this paper, we develop a systematic framework to study the dispersion surfaces of Schr{\"o}dinger operators $ -\Delta + V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is periodic with respect to a lattice $\Lambda \subset \mathbb{R}^n$ and respects the symmetries of $\Lambda$. Our analysis combines the theory of holomorphic families of operators of type (A) with the seminal work of Fefferman--Weinstein \cite{feffer12}. It allows us to extend results on the existence of spectral degeneracies past a perturbative regime. As an application, we describe the generic structure of some singularities in the band spectrum of Schr\"odinger operators invariant under the three-dimensional simple, body-centered and face-centered cubic lattices.